Part 19 - Mensuration formulae

The bhasya (p. 258) on Tattvarthadhigamasutra (III, 11) furnishes us with the following 6 formula:- (1) C = V 10 d² 5

5 In Anuyogadvarasutra (sutra 146, p. 235) the circumferece of a palya of 1 lac yojanas in diameter is given as under:- " se jahanamae palle sia egam joyanasayasahassam ayamavikakhambhenam tinni joyanasayasahassaim solasa sahasnai donni a sattavise joyanasae tinni a kose atthavisam ca dhanusayam terasa ya amgulai addham agulam ca kimci visesahiam parikkhevenam pannatte " In the commentary (p. 236) on this work by Maladharin Hemacandra, the following verse in Prakrit has been quoted:-

(3) c = V 4 h (d-h) (5) a = V 6 h² + cª Here C stands for the d and A for its area. (4) h = (d-V (d²-c² ) 2 (6) d = (h² +₂²). / h circumference of a circle of diameter The arc of a segment of a circle less than a semicircle, its chord and its height or arrow are denoted as a, c and h respectively. Over and above these 6 formula mentioned above, the bhasya (p. 258) gives us a rule as below: The portion of the circumference of a circle between (bounded by) two parallel chords is equal to half the difference between the corresponding arcs. In Ksetrasamasa of which the authorship is attributed to Umasvati, only the fourth formula is not to be found. Furthermore, the rule pertaining to finding out the arrow mentioned there can be expressed as /= V (a²-c²) This topic is dealt with by Ratnasekhara Suri in his Laghuksetrasamasa in the following hemistiches of the gathas 188-190:- 66 "vikkhambhavaggadahagunamulam vassa parirao hoi "" "viu supihutte caugunausugunie mulabhiha jiva "" "usuvaggi chaguni jivavaggajue mulam hoi ghanupittha "" Trilokasara, too, furnishes us with the formula here given and some more. All of them can be mentioned as under: " parihi ti lakkha solasa sahassa do ya saya sattavisa'hiya | kosatiya atthavisam dhanusaya teramgula'ddhahiyam || " [ paridhistrayo laksah sodasa sahasra dve ca sate saptavimsatyadhike | krosatrikamastavimsam dhanuhsatam trayodasangulani ardhadhikani || ] 1 Compare Ganitasarasangraha VII 43, 73, and Mahasiddhanta (Benares edn. XV, 90, 94, 95 of Aryabhatta. According to the Greek Heron of Alexandria (c. 20 v) a = V 4 h² + c²+1/4 or V 4 h² + (V 4 h² + c²-c) 1. The Chinese Hue who died in 1075 A. D. gives the formula as a = c x 2¹². l 2 In Ksetrasamasa (v. 7) as well as in the curni on Jambudvipaprajnapti, instead of mulam there is karani . 3-5 Sanskrit rendering: - viskambhava rgadasagunamulam vrttasya paridhirbhavati | bi (vigate ) suprthutve caturgunepugunite mulamiha jiva | isuvarga sadgune jivavargayute mulam bhavati dhanuhprstham |

(I) C (gross) = 3 d (2) C (subtle or neat) = V 10 d² (3) A = } C d 9 (4) r = s where r is the radius of a circle equivalent to a square of side s; thus T= 5) c² = 4 h (d-h) (6) a² = 6 h²+c² (7) d = c² + 4 h³ 4 h (8) A (gross) = V 10. c. (9) A (neat) = (10) d= c² + (2 h)² 4 h h xiao (c+h) h = (1/60) 2 (11) h = Va²-c² 6 (12) h = } (d-Vdc ) (13) d = 1/3 (g² -h ) 2 h (14) h = Va²+}a 2-d HAIGH (15) a² = 4 h (d+}) (16) c² = a²-6 h² Out of these formula, the 1 st three are given in gatha 311, the 4 th in 18, the 5 th and the 6 th in 760, the 7 th in 761, the 8 th and the 9 th in 762, the 10 th and the 11 th in 763, the 12 th in 764, the 13 th and the 14 th in 765, and the 15 th and the 16 th in 766. In Trilokasara (gatha 309) we find the discussion about the breadth of an annulus (valayavyasa) and the diameter of its edge (sucivyasa). Gommatasara supplies us with formula about volumes of a prism etc. For instance, from gatha 17 we learn that the volume of a prism = base x height. The gatha 19 furnishes us with two formula as under:- (i) Volume of a cone or a pyramid = 3 base x height. (ii) Volume of a sphere = (radius) 3 Gathas 22 and 23 lead us to the following conclusions:Volume of a conical shape = (Circumference ) ³ x height. This is on the supposition that the height equals (approximately) A circumference. The gatha 114 deals with an isosceles trapezium.

If a, b and h represent the face, the base and the altitude of an isosceles trapezium, we can have the following results:- (1) The rate of decrease of b or increase of a= (2) Area = (a+b)h. b-a h (3) At a height h' above the base, the breadth of the figure will be b-baland at a depth h" below the face, the breadth will be a+bh'. h It may not be amiss to mention that some geometrical figures are suggested in the following passage of Dasasruta. skandha (VII):- "masiyena bhikkhupadimam padipannassa anagarassa chavivaha goyaracariya pannatta, tamjaha- peda, apeda, gomuttiya, payamgavihiya, sambukkavaha ', gamtumpancagaya " Before I finish this section about the geometrical know. ledge of the Jainas, I may mention two problems. One is referred to in Bhagavati (sutras 726 and 727). It deals with the minimum number of pradesas (shots, literally spots) required to construct various geometrical forms. To give a clear idea, I may give a tabular form:Geometrical form *** Circle Sphere Triangle Triangular pyramid Square Cube Line 400 Minimum number of Minimum number of even shots odd shots 5... 7... 000 * 9 * 3... 000 : : 35... 9... 27... 500 006 0 0 0 I 2 009 32 0 0 0 . 644 co Rectangle 000 Parallelopiped 3... 15... 45... 0 0 0 I 2 252 6 The other problem deals with the different strata of Meru mountain and it is treated in Jambudvipaprajnapti. 1 Vide Uttaradhyayanaiutra (XXX, 19). 2 See Sthananga (VI; sutra 514). 3 Sanskrit rendering:- masikena bhiksupratimam pratipannasyanagarasya sada़vidha gocaracarya prasapta, tad yathaprela, ardhapela, gomutrika, patangavidhika sambukavartta, gatvapratyagata |